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# User:Frank Ellermann

## Contents

Please ignore the old T-Online mail address in sequences submitted by me years ago. In theory that could be fixed, but I won't bother the editors with minor nits while they are busy to create OEIS.org and this Wiki.

Today editing existing sequences is far simpler than it used to be about nine years ago based on the tricky internal format : Find the sequence, log in with your OeisWiki account, click edit, update say a broken link, optionally add an edit comment, click submit, and some minutes later I got an e-mail that the proposed modification was approved by an associate editor. Actually I found the how to after the fact.

## Known OEISwiki bugs and limitations

• Instant commons is a mechanism introduced in 2011 permitting MediaWiki projects to import resources from WikiMedia commons. It is documented on the MediaWiki wiki. If you see a red link instead of an image below it does not yet work here:
• Special:MyUploads (same idea as MyPage and MyTalk) shows uploads when the next MediaWiki version is installed here.
• At the moment subpages in the template namespace don't work here, this is arguably a missing MediaWiki default.
• <abbr title="abbreviation">ABBR</abbr> does not yet work here, use {{abbr|abbreviation|ABBR}} to bypass this limitation also for acronyms:
XHTML test: <abbr title="Encyclopædia of Integer Sequences">EIS</abbr>, template:abbr test: EIS.
• The OpenSearch descriptions for the OEIS wiki and the OEIS proper search link relations did not pass validation and in fact did not work with Chrome.
• The list of supported media types for Special:Upload does not yet include any video type, notably .ogv is not yet allowed. Maybe videos below 2MB could be encapsulated in the permitted .pdf or .gz types.
• The CC-BY-NC license used here is incompatible with the commons CC-BY-SA limits — whatever that means (IANAL), see my commons page hosted by Google.
• The short version of the OEIS movie available on YouTube cannot be embedded on OeisWiki pages, because
1. I got no permission to upload it as .ogv to commons due to the license incompatibility (see above),
2. without InstantCommons that would anyway not help here (see above), but might be nice on Wikipedia(s),
3. it also cannot be uploaded here — the 998KB are okay, but video MIME types are not allowed (see above),
4. the short version apparently does not yet exist on the OEIS proper (the pages above the OEIS wiki), and
5. even if it existed on OEIS proper the OEIS MediaWiki installation could consider it as external and refuse to treat it as embedded.
• As of September, 2011 the oldest secure legacy MediaWiki version was 1.16.5 (May) and the current version was 1.17 (June), anything older is insecure.
• Parser function {{nse:101}} should (roughly) work like {{anchorencode:{{ns:101}}}}. At the moment only the latter is supported here and gives Extras_talk, while the former fails with Template:Nse:101.
[talk]

## Sequence of the Day for April 19

A092287: $\prod_{j = 1}^n \prod_{k = 1}^n \gcd(j, k),\ n \ge 0.$

 { 1, 1, 2, 6, 96, 480, 414720, 2903040, ... }

Peter Bala conjectures that the order of the primes in the prime factorization of a(n) is given by the formula

$\operatorname{ord}_p\ a(n) = \sum_{k = 1}^{\lfloor \log_p(n) \rfloor} \left\lfloor\frac{n}{p^k}\right\rfloor ^2 = \left\lfloor \frac{n}{p} \right\rfloor ^2 + \left\lfloor \frac{n}{p^2} \right\rfloor ^2 + \left\lfloor \frac{n}{p^3} \right\rfloor ^2 + \cdots .$

Charles R Greathouse IV proved Bala's conjecture very recently.

Comparing this with the de Polignac–Legendre formula for the prime factorization of n!, i.e.

$\operatorname{ord}_p\ n! = \sum_{k = 1}^{\lfloor \log_p(n) \rfloor} \left\lfloor\frac{n}{p^k}\right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \cdots ,$

this suggests that a(n) can be considered as a generalization of the factorial numbers (the product between braces is obviously 1 if n is noncomposite)

$\frac{a(n)}{n!} = \left( \prod_{k = 1}^{n-1} \gcd(n, k) \right)^2 \frac{a(n-1)}{(n-1)!},\quad n \ge 1.$

Recurrence:

$a(0) := 1;\ a(n) := n \left( \prod_{k = 1}^{n-1} \gcd(n, k) \right)^2 a(n-1),\quad n \ge 1.$

Formula:

$a(n) = n! \left( \prod_{j = 1}^{n} \prod_{k = 1}^{j-1} \gcd(j, k) \right) ^2,\quad n \ge 0.$